This is further corroborated by
fig:gwline which shows the numerical and analytical
vertical velocity perturbation at \(z = 5\)  km for the two grids.
The vertical velocity perturbation for \(\Delta x = 3\)  km
in fig:gwcontours can be directly compared to
Figure 2 in [1]}.
With the same polynomial order and resolution
our entropy-stable DG scheme is clearly more accurate, which may
be surprising since [1]} used more accurate Gauss
quadrature.
The main reason for this is that, as proven in , the entropy stable scheme
is well-balanced for isothermal atmospheres.
It is well known that well-balanced schemes are better at approximating small
perturbations around balanced states [3]},
which is exactly the situation in this test problem.
<FIGURE>